Noncrystallographic Coxeter group H4 in E8 and quaternions

Abstract:
The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described by quaternions. Embedding the noncrystallographic Coxeter group H4 into the Weyl group W(E8) has been described using matrix generators in the icosian basis with an emphasis on the relevant Coxeter elements. The conjugacy classes of H4 in terms of quaternions and its maximal subgroups have been discussed in detail.